Explicit determinantal formulas for solutions to the generalized Sylvester quaternion matrix equation and its special cases

Abstract

Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore-Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer's rule) to the quaternion two-sided generalized Sylvester matrix equation ${\mathbf{A}}_1{\mathbf{X}}_1{\mathbf{B}}_1+{\mathbf{A}}_2{\mathbf{X}}_2{\mathbf{B}}_2=\mathbf{C}$ and its all special cases when its first term or both terms are one-sided. Finally, we derive determinantal representations of two like-Lyapunov equations.

References in corpus (1)

• Roth's solvability criteria for the matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$ over the skew field of quaternions with an involutive automorphism $q\mapsto \hat{q}$